吾人已於實驗室中製作完成一套球桿系統，本系統主要包括電樞控制直流馬達(armature controlled DC motor)、機台、鋼球、鋁桿、位置感測器(position-sensor)以及角度感測器(potentiometer)等等。
本論文以球桿系統為例探討控制上調節問題(regulation problem)，由於球桿系統存在離心力項，所以系統的相對階數(relative degree)沒有被適當的定義(not well defined)。然而此項提供強烈的正回授量，容易使系統產生跳動現象而呈現不穩定，若桿子角度控制不妥時，容易使球突然飛出而脫離桿上的滑軌，此時會得到不正確的球位置回授訊號，因而產生錯誤的回授控制。
文中，本人使用倒階 (backstepping)控制方法，其為遞迴程序重複選擇Lyapunov function的一種回授控制設計，且能在很少限制條件下時常解決穩定性、追蹤和強健控制上的問題。在系統控制器設計中利用高增益的控制參數，使離心力項在閉迴路的影響被衰減，且系統達到全域漸進穩定(GAS)。

A ball and beam system has been built. This system consists of an armature controlled DC motor, a machine platform, a ball, a beam, a position-sensor and a potentiometer.
This thesis considers the regulation problem of the ball and beam system. Due to existence of the centrifugal force, the system relative degree is not well defined. Moreover, the centrifugal force provides a strong positive feedback and easily leads to the peaking phenomenon. If the beam angle is not well controlled then the ball may fly off the beam. It will induce incorrect the feedback signal of the ball position, therefore produce error feedback control.
In this thesis, the backstepping control approach has been used to design a peaking-free ball and beam system. It is a recursive procedure that interlaced the choice of a Lyapunov function with the design of feedback control, and backstepping control often solves stabilization, tracking and robust control problems under less restrictive conditions. In this thesis, using high-gain control parameters, the effect of the centrifugal force on the closed-loop can be arbitrarily attenuated, and the system reachs global asymptotic stability.

[1] J. Hauser, S. Sastry and P. Kokotovic, “Nonlinear Control via Approximation Input-Output Linearization: The Ball and Beam Example,” IEEE Trans. on Automatic Control, Vol. 37, No. 3, pp. 392-398, 1992.
[2] P. Kokotovic, “The Joy of Feedback: Nonlinear and Adaptive,” IEEE Control Systems Magazine, Vol. 12, No. 3, pp. 7-17, 1992.
[3] C. Barbu, R. Sepulche, W. Lin and P. Kokotovic, “Global Asymptotic Stabilization of the Ball-and-Beam System,” Proceedings of the 36th IEEE Conference on Decision and Control, Vol. 3, pp. 2351-2355, 1997.
[4] 陳永隆，「可變結構理論在撓性臂控制器上之應用」，國立成功大學工程科學研究所碩士論文，民國八十四年六月。
[5] 辛俊光，「永磁式直流有刷馬達之參數自動鑑別系統」，國立成功大學航空太空工程研究所碩士論文，民國八十六年六月。
[6] S. R. Hebertt, “On the Control of The Ball and Beam System: A Trajectory Planning Approach,” Proceedings of the 39th IEEE Conference on Decision and Control, Vol. 4, pp. 4042-4047, 2000.
[7] J. J. E. Slotine and W. Li, “Applied Nonlinear Control,” Prentice-Hall, Englewood Cliffs, N. J., 1991.
[8] M. Krstic, I. Kanellakopoulos, and P. Kokotovic, “Nonlinear and Adaptive Control Design, ” Wiley-Interscience, New York, 1995.
[9] 陳慶全，「適應性Backstepping控制」，國立臺灣科技大學機械工程研究所碩士論文，民國八十九年八月。
[10] J. Zhao and I. Kanellakoulos, “Flexible Backstepping Design for Tracking and Disturbance Attenuation,” Int. J. Robust Nonlinear Control, pp. 331-348, 1998.
[11] 楊文吉，「倒單擺之製作及研究：甩上控制與修正式積分可變結構控制器之設計」，國立成功大學工程科學研究所碩士論文，民國八十六年六月。
[12] 張金城，「球桿系統動態平衡的控制」，國立臺灣科技大學工程技術研究所碩士論文，民國八十五年六月。
[13] H. K. Kim, D. H. Lee, T. Y. Kuc and T. C. Yi, “A Backstepping Design of Adaptive Robust Learning Controller for Fast Trajectory Tracking of Ball-Beam Dynamic Systems,” IEEE International Conference on Systems, Man, and Cybernetics, Vol. 3, pp. 2311-2314, 1996.